The Sandusky men admitted they made cash deposits of just under $10,000 on multiple occasions between 2007 and 2009 to avoid having to report the cash income from the family’s two Dianna’s Deli restaurants in Sandusky and from Dianna’s in Port Clinton. “Each of these defendants personally engaged in numerous structured transactions,” Joseph Wilson, an assistant U.S. attorney, told the court.

Each faces up to five years in prison and a $250,000 fine when they are sentenced Feb. 11, although Judge Helmick said he will have the option to place them on probation rather than send them to prison.

Also charged in the conspiracy are the men’s parents, Haralambos Gonos, 52, and Sofia Skoura, 52.

Haralambos Gonos, who is scheduled for a change-of-plea hearing in federal court Oct. 30, is charged with conspiracy to structure financial transactions to evade filing currency transaction reports and with conspiracy to structure financial transactions to evade filing Form 8300 — an Internal Revenue Service form required for cash payments of $10,000 or more — between March, 2010, and November, 2011.

Both Haralambos Gonos and Ms. Skoura also are charged with conspiracy to engage in bulk cash smuggling for allegedly taking cash in excess of $10,000 with them to Greece, where they have a home.

Mr. Wilson said after the hearing that prosecutors believe the Gonos family ran “essentially cash businesses,” though he did not say why.

“In some cases, people simply don’t want the government to know their business,” he said. “Some people can do it with the purpose of evading taxes.”

After the plea hearing ended, Judge Helmick found that Chris Gonos had violated the conditions of his bond by testing positive for marijuana use Sept. 25 and Oct. 10. The judge did not alter the terms of his release, but warned him that future violations could result in a change of his bond and could affect the sentence he imposes in February.